you should understand how the foundations of math work before doing advanced math
Is this merely something that set theoreticians believe, or do mathematicians that are experts at other branches of math actually find set theory useful for their work?
Can you in practice use set theory to discover something new in other branches or math, or does it merely provide a different (and less convenient) way to express things that were already discovered otherwise?
Many statements are undecidable in ZFC; what impact does that have on using set theory as a foundation for other branches of math?
Can you in practice use set theory to discover something new in other branches or math, or does it merely provide a different (and less convenient) way to express things that were already discovered otherwise?
The value of set theory as a foundation comes more from being a widely-agreed upon language that is also powerful enough to express pretty much everything mathematicians can think up, rather than as a tool for making new discoveries. I think it’s worth learning at least at a shallow level for this reason, if you want to learn advanced math.
I’d add that set theory gives you tools (like Zorn’s lemma and transfinite induction) that aren’t particularly exciting themselves, but you do need them to prove results elsewhere (e.g. Tychonoff’s theorem, or that every vector space has a basis).
That said there are some examples of results from formalizing math/ logic being used to prove nontrivial things elsewhere. My favourite example is that the compactness theorem of first order logic can be used to prove the Ax–Grothendieck theorem (which states that injective polynomials from C^n → C^n are bijective). I find this pretty cool.
I don’t know or think set theory is special. I just wanted to start at the very beginning. Another reason why I chose to start at set theory is because that is what Soares and Turntrout did and I just wanted somewhere to start (and I needed an easy-ish environment to level up in proofs). The foundations of math seemed like a good place. I plan to do linear algebra next because I think I need better linear algebra intuition for pretty much everything. It seems like it helps with a lot.
Is this merely something that set theoreticians believe, or do mathematicians that are experts at other branches of math actually find set theory useful for their work?
Can you in practice use set theory to discover something new in other branches or math, or does it merely provide a different (and less convenient) way to express things that were already discovered otherwise?
Many statements are undecidable in ZFC; what impact does that have on using set theory as a foundation for other branches of math?
The value of set theory as a foundation comes more from being a widely-agreed upon language that is also powerful enough to express pretty much everything mathematicians can think up, rather than as a tool for making new discoveries. I think it’s worth learning at least at a shallow level for this reason, if you want to learn advanced math.
I’d add that set theory gives you tools (like Zorn’s lemma and transfinite induction) that aren’t particularly exciting themselves, but you do need them to prove results elsewhere (e.g. Tychonoff’s theorem, or that every vector space has a basis).
That said there are some examples of results from formalizing math/ logic being used to prove nontrivial things elsewhere. My favourite example is that the compactness theorem of first order logic can be used to prove the Ax–Grothendieck theorem (which states that injective polynomials from C^n → C^n are bijective). I find this pretty cool.
I don’t know or think set theory is special. I just wanted to start at the very beginning. Another reason why I chose to start at set theory is because that is what Soares and Turntrout did and I just wanted somewhere to start (and I needed an easy-ish environment to level up in proofs). The foundations of math seemed like a good place. I plan to do linear algebra next because I think I need better linear algebra intuition for pretty much everything. It seems like it helps with a lot.